Global existence and smoothness for solutions of viscous Burgers equation. (2) The unbounded case: a characteristic flow study

We show that the homogeneous viscous Burgers equation $(\partial_t-\eta\Delta) u(t,x)+(u\cdot\nabla)u(t,x)=0,
(t,x)\in{$\mathbb{R}$}_+\times{$\mathbb{R}$}^d$ $(d\ge 1,\eta >0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(\mathrm{\mid }x\mathrm{\mid })$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $\eta >0$ using Schauder estimates. The result extends with few modifications to initial conditions growing abnormally large in regions with small relative volume, separated by well-behaved bulk regions, provided these are stable under the characteristic flow with high probability. We provide a large family of examples for which this loose criterion may be verified by hand.